where we recall that \(\ell _{c,2} = \sqrt{\sigma _{2a}/\rho _2 g}\)  is the capillary length related to interface (II).
This condition is analogous to the asymptotic matching condition with the static meniscus introduced by [1]} in the one-liquid dip-coating problem.
Combining equations (REF ) and (REF ), we arrive at the dimensional scaling expression
\(\delta h_{\infty } \sim \ell _{c,2} \, \left(-\tau _{12,\ast } \, \frac{ \ell _{c,2}}{\sigma _{2a}} \right)^2,\) 
